This paper presents a new explicit infinite family of 2-quasi-perfect $p$-ary Lee codes of length $\frac{q-1}{2}$ and dimension $\frac{q-1}{2}-2k$ for $q = p^k \ge 14$, $p\geq 5$ a prime. Our codes are derived from the generating set $H_q = \{(a, a^3) \mid a \in \mathbb{F}_q^*\}$ of the finite field $\mathbb{F}_{q^2}$. Furthermore, we bridge between 2-quasi-perfect Lee codes constructed by Mesnager, Tang, and Qi and well-known abelian Ramanujan graphs, specifically Li's graphs and finite Euclidean graphs, providing a unified theoretical framework for these families.

翻译：本文给出了一个显式无限族$2$-准完美$p$元Lee码的新构造，其长度为$\frac{q-1}{2}$，维数为$\frac{q-1}{2}-2k$，其中$q = p^k \ge 14$，$p\geq 5$为素数。我们的码源自有限域$\mathbb{F}_{q^2}$的生成集$H_q = \{(a, a^3) \mid a \in \mathbb{F}_q^*\}$。此外，我们建立了Mesnager、Tang和Qi构造的$2$-准完美Lee码与著名阿贝尔Ramanujan图（特别是Li图与有限欧几里得图）之间的联系，为这些族提供了统一的理论框架。